PROCEEDINGS OF SPIEdata.bettstetter.com/publications/okeeffe-2019... · story, the bio-inspired community has also mimicked the minimalism of Vicsek's and other models of swarming

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A review of swarmalators and theirpotential in bio-inspired computing

Kevin O'Keeffe, Christian Bettstetter

Kevin O'Keeffe, Christian Bettstetter, "A review of swarmalators and theirpotential in bio-inspired computing," Proc. SPIE 10982, Micro- andNanotechnology Sensors, Systems, and Applications XI, 109822E (13 May2019); doi: 10.1117/12.2518682

Event: SPIE Defense + Commercial Sensing, 2019, Baltimore, Maryland,United States

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A review of swarmalators and their potential in bio-inspiredcomputing

Kevin O’Keeffea and Christian Bettstetterb

aSenseable City Lab, Massachusetts Institute of Technology, Cambridge, MA 02139, UnitedStates.

bInstitute of Networked and Embedded Systems, University of Klagenfurt, 9020, Austria

ABSTRACT

From fireflies to heart cells, many systems in Nature show the remarkable ability to spontaneously fall intosynchrony. By imitating Nature’s success at self-synchronizing, scientists have designed cost-effective methodsto achieve synchrony in the lab, with applications ranging from wireless sensor networks to radio transmission.A similar story has occurred in the study of swarms, where inspiration from the behavior flocks of birds andschools of fish has led to low-footprint algorithms for multi-robot systems. Here, we continue this ‘bio-inspired’tradition, by speculating on the technological benefit of fusing swarming with synchronization. The subject ofrecent theoretical work, minimal models of so-called ‘swarmalator’ systems exhibit rich spatiotemporal patterns,hinting at utility in ‘bottom-up’ robotic swarms. We review the theoretical work on swarmalators, identifypossible realizations in Nature, and discuss their potential applications in technology.

Keywords: Synchronization, swarming, bio-inspired computing.

1. INTRODUCTION

In 1967 Winfree proposed a model for coupled oscillators that spontaneously synchronize.1 Beyond a criticalcoupling strength, the oscillators overcome the disordering effect of the dissimilarities in their natural frequenciesand spontaneously lock their cycles. Kuramoto later simplified Winfree’s model and solved it exactly.2 Sincethen, the study of sync has matured into a vibrant field.3–5 On the theoretical side, theorists have twistedKuramoto’s model in various ways resulting in rich phenomena like glassy behavior6–9 and chimeras states.10–12

On the applied side, coupled oscillators have found use in neurobiology,13–16 cardiac dynamics,17–19 and thebunching of school buses.20 An interesting application of sync is the design of ‘bio-inspired’ algorithms. Here,by aping the simplicity of coupled oscillator models, researchers have devised resource-efficient algorithms toprocure synchrony in the lab, useful for networked computing and robotics.21–24

A story with many parallels to the sync story has evolved in the study of swarms. In 1995 Vicsek proposed asimple model of swarming agents,25 which – like the sync transition of coupled oscillators – showed a transitionfrom disorder to order: beyond a critical coupling strength, the agents switched from a gas-like, incoherentstate to one in which the agents moved as a coherent flock. The novelty of the out-of-equilibrium nature ofthe flocking transition – the system is out of equilibrium since agents constantly consume energy to propelthemselves – piqued the minds of physicists and other theorists, which in turn helped give rise to the field ofactive matter,26–28 a field perhaps more vibrant than the field of synchronization. In a final mirroring of the syncstory, the bio-inspired community has also mimicked the minimalism of Vicsek’s and other models of swarmingto design novel algorithms for optimization29–34 and robotic swarms.33,35–38

These stories demonstrate swarming and synchronization are intimately related. In a sense, the two effectsare ‘spatiotemporal opposites’: in synchronization the units self-organize in time but not in space; in swarmingthe units self-organize in space but not in time. Given this conceptual twinship, it is natural to wonder aboutthe possibility of units which can self-organize in both space and time – that is, to wonder how swarmingand synchronization might interact. And more pertinent to this work, to wonder how a mix of swarming andsynchronization could be useful in technology.

Further author information: (Send correspondence to Kevin O’Keeffe; E-mail: [email protected])

Invited Paper

Micro- and Nanotechnology Sensors, Systems, and Applications XI, edited by Thomas George, M. Saif Islam, Proc. of SPIE Vol. 10982, 109822E · © 2019 SPIE

CCC code: 0277-786X/19/$18 · doi: 10.1117/12.2518682

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Several researchers have started to address these questions by analyzing systems that both sync and swarm.Von Brecht and Uminsky have generalized swarming particles by endowing them with an internal polarizationvector,39 equivalent to an oscillator’s phase. Others have attacked the problem from the other direction, byconsidering synchronizing oscillators able to move around in space.40–43 In these studies, however, the oscillators’movements affect their phase dynamics, but not the other way around; thus, the interaction between swarmingand synchronization is only one-way. Two-way interaction between swarming and synchronization has also beenconsidered: The pioneering work is the Iwasa-Tanaka model of chemotactic oscillators,44,45 i.e., oscillators whichinteract through a background diffusing chemical. More recent works were carried out by Starnini et al,46 Belovset al,47 and O’Keeffe et al48 who proposed ‘bottom-up’ toy models without reference to a background medium.O’Keeffe et al called the elements of their systems ‘swarmalators’, to capture their twin identities as swarmingoscillators, and to distinguish them from the mobile oscillators mentioned above for which the coupling betweenswarming and synchronization is unidirectional.

This paper reviews research on swarmalators and other systems that mix swarming and synchronization. Ourmain motivation is to identify if the interplay of sync and swarming can be useful for bio-inspired computingand related engineering domains. We outline the theoretical work on swarmalators, experimental realizations,and finally conjecture on their technological utility.

2. MODELS COMBINING SWARMING AND SYNCHRONIZATION

To combine synchronization with swarming, we first define what we mean by swarming. While to our knowledgethere is no universally agreed on definition, swarming systems typically have at least one of two key features:(i) aggregation, arising from a balance between the units’ mutual attraction and repulsion and (ii) alignment,referring to the units’ tendency to align their orientation in space and move in a flock. Synchronization can thusbe combined with either aggregation alone, alignment alone, or with both aggregation and alignment. In whatfollows, we present classes of model based on these three ways to combine sync and swarming.

2.1 Aggregation and Synchronization

The paradigmatic model of biological aggregation has the form

xi =1

N

N∑j 6=i

Iatt(xj − xi)− Irep(xj − xi), (1)

where xi ∈ Rd (usually with d ≤ 3) is the i-th particle’s position, Iatt captures the attraction between particles,and Irep captures the repulsion between them. The competition between Iatt and Irep gives rise to congregationsof particles with sharp boundaries, in accordance with many biological systems (see49,50 for a review).

The paradigmatic model in synchronization is the Kuramoto model:51

θi = ωi +K

N

N∑j 6=i

sin(θj − θi). (2)

Here θi ∈ S1 and ωi are the phase and natural frequency of the i-th oscillator. The sine term captures theoscillators’ coupling, where K > 0 means oscillators tend to synchronize, and K < 0 means oscillators tendto desynchronize. As mentioned, beyond a critical coupling strength Kc, a fraction of oscillators overcome thedisordering effects imposed by their distributed natural frequencies ωi and spontaneously synchronize.

To study the co-action of synchronization and aggregation, a natural strategy would be to stitch the aggre-gation model (1) and the Kuramoto model (2) together. This was the approach taken by O’Keeffe et al48 whoproposed the following swarmalator model:

xi =1

N

N∑j 6=i

[xj − xi|xj − xi|

(1 + J cos(θj − θi)

)− xj − xi|xj − xi|2

](3)

θi = ωi +K

N

N∑j 6=i

sin(θj − θi)|xj − xi|

. (4)


Figure 1. Swarmalator states. Scatter plots in the (x, y) plane, where the swarmalators are colored according to theirphase. (a) Static sync for (J,K) = (0.1, 1). (b) Static async (J,K) = (0.1,−1). (c) Static phase wave (J,K) = (1, 0). (d)Splintered phase wave (J,K) = (1,−0.1). (e) Active phase wave (J,K) = (1,−0.75).

Equation (3) models phase-dependent aggregation and Equation (4) models position-dependent synchronization.The interaction between the space and phase dynamics is captured by the term 1 +J cos(θj − θi). If J > 0, “likeattracts like”: swarmalators are preferentially attracted to other swarmalators with the same phase, while J < 0indicates the opposite. For simplicity the authors considered identical swarmalators ωi = ω, and by a change ofreference they set ω = 0.

The swarmalator model exhibits five long-term collective states. Figure 1 showcases these states as scatterplots in the (x, y) plane, where swarmalators are represented by dots and the color of each dot represents theswarmalator’s phase θ (color, recall, can be mapped to S1 and so can be used to represent swarmalators’ phases).The parameter dependence of these states are encapsulated in the phase diagram shown in Figure 1. The firstthree states, named the static sync, static async, and static phase wave, are – as their names suggest – staticin the sense that the individual swarmalators are stationary in both position and phase. This stationarityallows the density of these swarmalators ρ(x, θ) in these states to be constructed explicitly (the density ρ(x, θ)is interpretted in the Eulerian sense, so that ρ(x + dx, θ + dθ) gives the fraction of swarmalators with positionsbetween x and x+dx and phases between θ and θ+dθ). In the remaining splintered phase wave and active phasewaves states swarmalators are no longer stationary. In the splintered phase wave state swarmalators executesmall oscillation in both space and phase within each cluster. In the active phase wave the swarmalators splitinto counter-rotating groups – in both space and phase – so that 〈θi〉 = 〈φi〉 = 0, where φi = arctan(yi/xi) is thespatial angle of the i-th oscillator and angle brackets denote population averages. The conservation of these twoquantities follows from the governing equations; the pairwise terms are odd and thus cancel under summation.Three-dimensional analogues of the five collective states were also reported.

The stability properties of the static async state are unusual. Via a linear stability analysis in density space,an integral equation for the eigenvalues λ was derived. Yet numerical solutions to this integral equation, inthe parameter regime where the static async state should be stable, produced a leading eigenvalue so small inmagnitude that its sign could not be determined reliably. Thus, the stability of the state could not be analyticallyconfirmed. There is however a parameter value at which the magnitude of λ increases sharply, which was usedto derive a pseudo critical parameter value Kc ≈ −1.2J marking the apparent (i.e. as observed in simulations)destabilization of the state. Nevertheless, the true stability of the static async state remains a puzzle.

Some extensions of the work in48 have been carried out. One addresses the conspicuously empty upper rightquadrant in the (J,K) parameter plane in Figure 2, where only the trivial static sync state appears. By addingphase noise to (4), Hong discovered52 the active phase wave exists for K > 0. The splintered phase wave howeverwas not observed. O’Keeffe, Evers, and Kolokolnikov53 have extended (3) by allowing phase similarity to affectthe spatial repulsion term, as well as the spatial attraction term (i.e. they multiply the second term in (3) by aterm 1 + J2 cos(θj − θi)). This led to the emergence of ring states, an example of which is depicted in Figure 3.They constructed and analyzed the stability of these ring states explicitly for a population of given size N .Analytic results for arbitrary N are potentially useful for robotic swarms, which presumably are realized in thesmall N limit. Another offshoot of this analysis was a heuristic method to predict the number of clusters whichform in the splintered phase wave state; they viewed each cluster of synchronized swarmalators as one giantswarmalator which let them re-imagine the splintered phase wave state as a ring, allowing them to leverage theiranalysis. A precise description of the number of clusters formed is an open problem.


-1.0 -0.8 -0.6 -0.4 -0.2 0.2 K-0.2

0.2

0.4

0.6

0.8

1.0J

Static async

Static async

Active

phase wave

Splintered

phase wave

Static

sync

Static

phase wave

Static

sync

Figure 2. Phase diagram. Locations of states of the model defined by equations (3) and (4) in the (J,K) plane. Thestraight line separating the static async and active phase wave states is a semi-analytic approximation given by J ≈ 1.2K– see Eq. (18) in.48 Black and red dots show simulation data. The red dashed line simply connects the red dots and wasincluded to make the boundary visually clearer. Note, this figure has been reproduced from.48

-0.8-0.6-0.4-0.2 0.2 0.4 0.6x

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6y

Figure 3. Ring phase state. Swarmalators are represented by colored dots in the (x, y) plane where the color of eachdot represents its phase. The state is found by numerically integrating the governing equations in53 with (J1, J2,K,N) =(0, 0.8, 0, 100).

Iwasa-Tanaka model. Iwasa and Tanaka proposed and studied a ‘swarm-oscillator’ model in a series ofpapers.44,54–57 The inspiration for their work comes from chemotactic oscillators, i.e., oscillators moving aroundin a diffusing chemical which mediates their interactions. They began with the general model

Xi(t) = f(Xi) + kg(S(ri, t)) (5)

mri(t) = −γri − σ(Xi)S (6)

τ∂τS(r, t) = −S + d∂2S +∑i

h(X)δ(r− ri), (7)

where Xi represents the internal state (which will later be identified as a phase), r represents the position of thei-th oscillator, and S represents the concentration of the background chemical. By means of a center manifold

calculation and a phase reduction technique they derived the simpler equations


ψi(t) =∑j 6=i

e−|Rji| sin(Ψji − α|Rji| − c1) (8)

ri(t) = c3∑j 6=i

Rjie−|Rji| sin(Ψji − α|Rji| − c2) (9)

where Rji = Rj −Ri, Ψji = ψj − ψi, and ψi is the i-th oscillator’s phase. We call this Iwasa-Tanaka model.Notice the space-phase coupling in this model is somewhat peculiar; in contrast to the swarmalator model givenby (3) and (4) the relative position Rji and relative phase Φji appear inside the sine terms in both the ri and

ψi equations. Another difference between the two models is that ri in (8) has no hardshell repulsion term, whichmeans the oscillators can occupy the same position in space.

The Iwasa-Tanaka model has rich collective states. An exhaustive catalogue of these states with respect tothe model’s four parameters is an ongoing effort.57 Highlights include a family of clustered states44,45,56 in whichswarmalators collect in synchronous groups. The spatial distributions of these groups depend on their phase,similar to the splintered phase wave (Figure 1). The authors speculate this phase clustering is reminiscent of thechemotactic cell sorting during biological development.44 The Iwasa-Tanaka model also produces ring states,45

as well as an interesting ‘juggling’ state54 in which the population forms a “rotating triangular structure whosecorers appear to ‘catch’ and ‘throw’ individual elements” – in other words, the population juggles the elementsaround the corners of a triangle (see Figure 1 in54). Aside from theoretical novelty, this juggling could conceivablybe exploited in robotic swarms, potentially allowing some form of relay between the elements.

2.2 Alignment and synchronization

In our proposed taxonomy, systems that combine alignment and synchronization are characterized by an internalphase θ and an orientation β without a dynamic spatial degree of freedom. In other words, the particles’ positionx might affect their θ and β dynamics, but x itself does not evolve in time. Although units characterized by justa phase θ and orientation β might seem odd, Leon and Liverpool studied a system with units which meet thesecriteria: a class of soft active fluids which constitute a ‘new type of nonequilibrium soft matter – a space-timeliquid crystal’.58 They developed a phenomenological theory of these soft fluids which allowed them to derivedynamical equations for order parameters quantifying the orientation order, phase order, and orientation-phaseorder. These revealed collective states which maximize each of these order parameters: aligned states withorientational order but no phase order, sync’d states with phase order but no orientational order, and states withboth phase and orientational order. They were able to partially analyze these states, and conjectured the statescould be realized in protein-filaments mixtures, such as cell cytoskeletons59,60 or tissue-forming cells.61

2.3 Alignment, aggregation, and synchronization

Systems with units that align, aggregate, and synchronize – and therefore have dynamic state variables x, θ, β –are the least well studied. A small study was carried out in,48 with the aim of checking if the swarmalator statesreported are generic, i.e., robust to the inclusion of alignment dynamics (it was found they were). Beyond thispreliminary study, the space of possible behaviors arising from alignment, aggregation, and synchronization islargely unexplored.

2.4 Alignment and aggregation

While not strictly within our proposed taxonomy of swarmalator systems, we give a brief review of studies onaggregation and alignment. We do this because alignment can be viewed as a type of synchronization, whereinstead of units adjusting to a common phase θ, units instead adjust a common orientation β. Or put anotherway, because an internal phase θ and orientation β are formally equivalent – both being circular variables –particles aligning their orientations is analogous to oscillators aligning their phases.

Vicsek set the paradigm of this class of models with a beautiful and simple model:

ri(t+ δt) = ri(t) + (νδt)n (10)

βi(t+ δt) = 〈βj〉|rj−ri|<r + ηi(t), (11)


where n = (cos(β), sin(β)), ηi(t) is a white noise, ν is the particles’ speed, and r is the coupling range. Forsufficiently strong noise this model exhibits a flocking transition, where particles switch from erratic incoherentmovements to moving in a unidirectional flock.

Leibchen and Levis62,63 modified the Vicsek model as follows:

ri = ν~ni (12)

βi = ωi +K

πR2

N∑j∈∂i

sin(θj − θi) +√

2Dηi, (13)

where ∂i is the set of neighbours of the i-th particle, K is the phase coupling, and D is the noise strength.The key new feature is the ωi term, signifying the particles have an intrinsic rotation. They find three collectivestates: a disordered state in which neither spatial order nor phase order exists, a chiral state in which macroscopic‘droplets’ of synchronized oscillators form in a sea of otherwise desynchronized oscillators, and finally a mutualflocking state in which oscillators with ‘opposite chirality cooperate to move coherently at a relative angle,forming non-rotating flocks’. The theoretical novelty of the latter state is that – in contrast to regular, non-moving, locally-coupled oscillators coupled in two or three dimensions – long-range synchronous clusters areobserved, despite the fact that the oscillators are only locally coupled.

3. SWARMALATORS IN THE REAL-WORLD

To our knowledge, there are just two works which unequivocally realize swarmalators in the real world – by thiswe mean, precisely, a real-world system in which a bidirectional space-phase coupling is unambiguously exhibited.These works are:

Swarmalatorbots. Bettstetter et al first realized the collective states of the swarmalator model (Figure 1)in the lab using small robots.64 They programmed ‘swarmalatorbots’ whose governing equations were derivedfrom the swarmalator model (3), (4).

Magnetic domain walls. To our knowledge, the first realization of a natural swarmalator system was foundby Hrabec et al when studying the magnetic domain walls.65 Ordinarily, domain walls are described by a singlespatial degree of freedom x. But as the authors note, the dynamics on the walls also depend on their internalstructure. The authors minimally describe this internal state by a one dimensional polarization angle of theinternal magnetic field θ. This allows the wall to be viewed as a point particle with position x and phase θ – tobe viewed as a swarmalator.65 An experimental study of coupling between two domain walls – usually just onewall is studied – revealed the walls can synchronize, which in turn affects the walls’ velocity. Richer space-phasedynamics, such as families of Lissajous curves, are also reported.

***

Beyond swarmalatorbots and magnetic domain walls, there are many systems in which a bidirectional couplingbetween swarming and synchronization might exist. We list these candidate swarmalator systems below:

Myxobacteria. Myxobacteria are bacteria commonly found in soil, and can produce interesting collectiveeffects.66 The phase variable of myxobacteria characterizes the internal growth cycle of the cell. Groups of cellsinteract through cell-to-cell contact, theorized to provide a channel through which swarming and synchronizationcan couple. Populations of myxobacteria can exhibit a ‘ripple phase’, in which complex patterns of wavespropagate through population.66 In contrast to other wave phenomena in biological systems, such as the well-studied Dictyostelium discoideum, these rippling waves do not annihilate on collision. In66 a Fokker-Planck typeequation was used to analyze rippling waves. Realizing them in a microscopic swarmalator system is an openproblem.

Biological microswimmers. ‘Microswimmers’ is an umbrella term for self-propelled microorganisms con-fined to fluids, such as celia, bacteria, and sperm.67 Groups of microswimmers show swarming behavior as aresult of cooperative goal seeking, such as searching for food or light. They can also synchronize: Here the


phase variable is associated with the rhythmic beating of swimmers’ tails which – through hydrodynamic in-teractions – can synchronize with the beatings of others swimmers’ tails. Whether or not this hydrodynamicsprovides a bidirectional coupling between sync and swarming – as required of swarmalators – is unclear, but tous seems plausible. Researchers have developed models of sperm under this assumption68 and found clustersof synchronized sperm consistent with real data.69 Vortex arrays of sperm have also been reported.70 Here,sperm self-organize into subgroups arranged in a lattice; within each subgroup, the sperm move in a vortex,wherein a correlation between the their angular velocity and their phase velocity is realized, reminiscent of thesplintered phase wave state (Figure 1). Simulations of realistic models have been carried out47 which show otherinteresting phenomena.

Japanese tree frogs. During mating season, male Japanese tree frogs attract females by croaking rhythmi-cally. The croaking of neighbouring frogs tend to anti-synchronize due to a precedence effect: croaking shortlyafter a rival makes a frog look less dominant. Researchers have theorized that this competition leads to mu-tual interaction between frogs’ space and phase dynamics, coupling sync and swarming. Models based on theseassumption produce ring-like states where frogs arrange themselves on the borders of fields with interestingspace-phase patterns, some of which are consistent with data on real tree-frogs’ behavior collected in the wild.

Magnetic colloids. Similar to magnetic domain walls, a colloid’s constituent particles can synchronize theorientation of their magnetic dipole vector when sufficiently close. When in solution, they are free to move aroundin space, creating a feedback loop between their space and phase dynamics. In ferromagnetic colloids confinedto liquid-liquid interfaces, Snezhko and Aranson71 found this interaction leads to the formation of ‘asters’ –star-like arrangements of particles whose spatial angles correlate with the orientations of their magnetic dipolevectors, equivalent to the static phase wave (Figure 1). Yan et al explored how synchronization is useful incolloids of Janus particles.72 Janus particles are micrometer-sized spheres with one hemisphere covered withnickle which gives them non-standard magnetic properties. In particular when subject to a precessing magneticfield, they oscillate about their centers of mass. This oscillation creates a coupling between particles, giving rise to‘synchronization-selected’ self-assembly. For example, zig-zag chains of particles and microtubes of synchronizedparticles can be realized.

4. SWARMALATORS IN BIO-INSPIRED COMPUTING

The computer engineering community has been inspired by coupled oscillators to develop new techniques forsynchronization in communication and sensor networks.21–24,73,74 Here, instead of Kuramoto-type oscillators(2), models of pulse-coupled oscillators have been borrowed. As the name suggests, pulse-coupled oscillatorscommunicate by exchanging short signals; this time-discrete variant of coupling is a more natural fit for appli-cation in technology, where smooth, continuous coupling – as exemplified by Kuramoto oscillators – is costly toachieve. The canonical model of pulse-coupled oscillators is the Peskin model,17 defined by

xi = S0 − γxi (14)

with S0, γ > 0 and xi is a voltage like state variable for the i-th oscillator. When the oscillators’ voltage reach athreshold value they (i) fire a pulse which instantaneously raises the voltage of all the other oscillators (ii) resettheir voltage to zero, along with any other oscillator whose voltage was raised above the threshold on accountof receiving a pulse. The synchronization properties of the Peskin model are well-studied.75,76 More recently,estimates for the convergence speed have been derived.77,78

The simplicity, distributed nature, adaptability, and scalability of pulse-coupled oscillators make them attrac-tive from an engineering point of view, where temporal coordination is often a goal. For example, synchronizationis required for many tasks in different layers of computing and communications systems79 such as in the align-ment of transmission slots for efficient medium access,80 scheduling of sleep cycles for energy efficiency,81 andcoordination of sensor readings to capture a scene from different perspectives.82 It is the relative synchronyof the networked units, not necessarily the absolute time, that is relevant here. The synchronization precisionrequired is determined by the specific task. The achievable precision depends on the environment and hardware,and is influenced by deviations in delays and phase rates. Experiments in laboratory environments show that theprecision is in the order of hundred microseconds with low-cost programmable sensor platforms21,83 and a few


microseconds with field-programmable gate array (FPGA)-based radio boards.84 Further use cases can be foundin acoustics (synchronizing multiple loudspeakers) and energy systems (synchronizing decentralized grids85,86),to give two examples.

Synchronization is also important in robotics in order to perform coordinated movements – and this is wheretemporal and spatial coordination unite. As mentioned, Bettstetter and colleagues64 extended the swarmalatormodel48 so that it could be applied to mobile robots. They implemented the extended model in the RobotOperating System 2 (ROS 2) and experimentally demonstrated that the space-time patterns achieved in the-ory (Figure 1) can be reproduced in the real world. Beyond realizing these specific states, we conjecture thatswarmalator-type models will enable novel self-assembly procedures in other robotics groups, in turn enablingcollaborative actions in monitoring, exploration, and manipulation. As envisioned in,64 underwater robots couldbe designed, which – in imitation of biological microswimmers – could both swim in formation and synchronizetheir fin movements, thereby enhancing their functional capabilities.

More speculatively, the swarmalator concept could be useful in the design of autonomous transport systems;for example, when multiple vehicles driving in a convoy have to avoid collisions with other convoys due to crossingsand for the purpose of overtaking. The model could also enable self-configuring distributed antenna arrays (orloudspeakers) in which multiple antenna elements (or sound sources) automatically arrange their positions andorientations to create specific radiation patterns used to send radio (audio) signals in a synchronous way. Anotherpromising application field is the planning and replanning of processes in factories, where products and machinesmust follow a certain space-time order and where standard optimization methods reach their limits due to thehigh system complexity. A final, more playful, application is in art: Artistic aerial light shows created by droneswarms running the swarmalator model produce charming visual displays.

5. DISCUSSION

The implicit strategy in the theoretical works we reviewed is: given a model, which spatiotemporal patternsemerge? Future work could consider the inverse strategy: given a desired spatiotemporal pattern, which formshould the model take on? – a question often considered in engineering contexts. From a more general perspective,one has to design the local rules and interactions that guide a self-organizing system toward a desired globalstate.87 Von Brecht et al have studied this inverse question for swarming systems;88 perhaps their tools couldbe extended to swarmalators.

Future work could also extend the reviewed models, which – recall – were designed to be as minimal as possible.A wealth of synchronization phenomena have been found by adorning the Kuramoto model with new featureslike mixed-sign coupling,89–92 non-local coupling,11,12 and delayed interactions.93 Equipping swarmalators withthese features would likely cure the poverty of phenomena in the first, third, and fourth quadrants of the (J,K)plane (Figure 2) and produce new dynamics in Iswasa-tanaka and Vicesek type models, too. Phase models moresophisticated than the Kuramoto model could also be explored; the Winfree model94 or the newly introducedJanus oscillators95 would be exciting to experiment with. Swarmalators with discrete and spatially local couplingalso merit study and are useful in computer and communication systems. Stitching the Peskin model (14) to theaggregation equation (1) or the Vicsek model (11) seems the natural way to do this. Continuing to study finitesystems of swarmalators53 is also pertinent, since many robotic systems lie in the low N regime. And finally,beyond richer phase models, the spatial dynamics of swarmalators could also be generalized. Even within theframework of the aggregation equation (1), a menagerie of spatial patterns have been catalogued.96 Swarmalatorcounterparts to these states would be exciting to explore.

We hope to have outlined that swarmalators have great potential in computing and other fields of tech-nology. Here we see swarmalators only as a case study in a broader class of systems with large technologicalutility: systems whose units have both spatial and internal degrees of freedom. The one-dimensional phase θ ofa swarmalator is perhaps the simplest instance of an internal degree of freedom, which can more generally berepresented by a feature vector f . This vector could describe a particle’s (three dimensional) magnetic or electricfield; a person’s political affiliation, mood, or health state; a bacteria’s phase in a non-circular developmentalcycle; and the mode of a robot, machine, or vehicle. The collective states arising when particles’ features f andpositions x defines a wide, scarcely investigated area. We anticipate an exploration of this area will bear rich

fruit in both Nature and technology.


Acknowledgements

The work of C. Bettstetter is supported by the Austrian Science Fund (FWF) [grant P30012] and the PopperKolleg Networked Autonomous Aerial Vehicles.

The work of Kevin O’Keeffe thanks Allianz, Amsterdam Institute for Advanced Metropolitan Solutions, Brose,Cisco, Ericsson, Fraunhofer Institute, Liberty Mutual Institute, Kuwait-MIT Center for Natural Resources andthe Environment, Shenzhen, Singapore- MIT Alliance for Research and Technology (SMART), UBER, VitoriaState Government, Volkswagen Group America, and all the members of the MIT Senseable City Lab Consortiumfor supporting this research.

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PROCEEDINGS OF SPIEdata.bettstetter.com/publications/okeeffe-2019... · story, the bio-inspired community has also mimicked the minimalism of Vicsek's and other models of swarming